256 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
Substituting these values, but retaining {a, b, c, f, g, h) as standing for their values 
a = 2\V, &c., the equation in 0 is found to contain the cubic factor 2X0 3 — 3 Yd 2 + Z, 
where it is to be observed that this factor equated to zero determines the values of 
0 which correspond to the points of contact with the cuspidal cubic of the tangents 
from the point (X, Y, Z), which is the intersection of the lines \x + gy + vz = 0, and 
\'x + gy + v'z = 0; and omitting the cubic factor, the residual equation is found to be 
( 
2cA" 
— 12c F 
-8/X 
-20gX 
-10 bX 
- 40AX 
- 20aX 
+ 15aF 
+ 5 hZ 
+ aZ 
-12fY 
+ 3 gY 
- 8bY 
+ mr 
+ 4 bZ 
+ icZ 
+ 7 gZ 
$0, l) 9 -0, 
where the form of the coefficients may be modified by means of the identical equations 
aJi. hY-f* gZ = 0, 
hX + bY +fZ = 0, 
gX 4- fY 4 cZ =0. 
The equation is of the 9th order, and there are consequently 9 conics. 
Annex No. 6 (referred to, No. 48).—Containing, with the variation referred to in the 
text, Zeuthen’s forms for the characteristics of the conics which satisfy four conditions. 
(1) 
( : : ) = n + 2m, 
( .'./) = + 4nn, 
( ://) = 4m + 4m, 
( • /¡I ) = 4n + 2m, 
( I III) = 2n + m ; 
(1,1) 
( .•. ) = 2m ( m+ n — 3) + t, 
( : / ) = 2in ( m + 2n — 5) + 2 t, 
( -//)== 2n (2m + n — 5) 4- 28, 
(///) = 2n ( m 4- n — 3) + 8; 
a i) 
( : ) = % [2m 3 + 6m 2 n — n 3 — 30m~ — 18mw 4 13n 2 4- 84?m — 42m 4- (6w + 3m — 26) t], 
( • / ) = £[(w4m)(—(m+w) 2 —7 (Mi4-M)4-48)+4mM(3?M4-3M — 13) + 2(3?n+3?i — 20)(&4t)], 
( // ) = [— mi 3 4 Gnm 2 + 2m 3 4- 13m 2 — 18tmm — 30m 2 — 42mî + 84m + (3m 4- 6m — 26) 8] ;